Ordinary Differential Equations/Successive Approximations

$y'=f(x,y)$ has a solution $y$ satisfying the initial condition $y(x_{0})=y_{0}$, then it must satisfy the following integral equation:

$y=y_{0}+\int _{x_{0}}^{x}f(t,y(t))dt$

Now we will solve this equation by the method of successive approximations.

Define $y_{1}$ as:

$y_{1}=y_{0}+\int _{x_{0}}^{x}f(t,y_{0})dt$

And define $y_{n}$ as

$y_{n}=y_{0}+\int _{x_{0}}^{x}f(t,y_{n-1})dt$

We will now prove that:

1. If $f(x,y)$ is bounded and the Lipschitz condition is satisfied, then the sequence of functions converges to a continuous function
2. This function satisfies the differential equation
3. This is the unique solution to this differential equation with the given initial condition.

Proof

First, we prove that $y_{n}$ lies in the box, meaning that $|y_{n}(x)-y_{0}|<{\frac {1}{2}}h$. We prove this by induction. First, it is obvious that $|y_{1}(x)-y_{0}|\leq {\frac {1}{2}}h$. Now suppose that $|y_{n-1}(x)-y_{0}|\leq {\frac {1}{2}}h$. Then $|f(t,y_{n-1}(t))|\leq M$ so that

$|y_{n}(x)-y_{0}|\leq \int _{x_{0}}^{x}|f(t,y_{n-1}(t))|dt\leq M(x-x_{0})\leq {\frac {1}{2}}Mw\leq {\frac {1}{2}}h$. This proves the case when $x_{0}<x$, and the case when $x<x_{0}$ is proven similarily.

We will now prove by induction that $|y_{n}(x)-y_{n-1}(x)|<{\frac {MK^{n-1}}{n!}}(x-x_{0})^{n}$. First, it is obvious that $|y_{1}(x)-y_{0}|<M(x-x_{0})$. Now suppose that it is true up to n-1. Then

$|y_{n}(x)-y_{n-1}(x)|\leq \int _{x_{0}}^{x}|f(t,y_{n-1}(t))-f(t,y_{n-2}(t))|dt<\int _{x_{0}}^{x}K|y_{n-1}(t)-y_{n-2}(t)|dt$ due to the Lipschitz condition.

Now,

$|y_{n}(x)-y_{n-1}(x)|<{\frac {MK^{n-1}}{(n-1)!}}\int _{x_{0}}^{x}||u-x_{0}|^{n-1}du={\frac {MK^{n-1}}{n!}}|x-x_{0}|^{n}$.

Therefore, the series of series $y_{0}+\sum _{n=1}^{\infty }(y_{n}(x)-y_{n-1}(x))$ is absolutely and uniformly convergent for $|x-x_{0}|\leq {\frac {1}{2}}w$ because it is less than the exponential function.

Therefore, the limit function $y(x)=y_{0}+\sum _{n=1}^{\infty }(y_{n}(x)-y_{n-1}(x))=\lim _{n\rightarrow \infty }y_{n}(x)$ exists and is a continuous function for $|x-x_{0}|\leq {\frac {1}{2}}w$.

Now we will prove that this limit function satisfies the differential equation.